Amit Ghosh

A conjecture for the sixth power moment of the Riemann zeta-function

The authors conjecture an asymptotic expression for the sixth power moment of the Riemann zeta function. They establish related results on the asymptotics of the zeta function that support the conjecture.

Nodal Domains of Maass Forms I

This paper deals with some questions that have received a lot of attention since they were raised by Hejhal and Rackner in their 1992 numerical computations of Maass forms. We establish sharp upper and lower bounds for the L2-restrictions of these forms to certain curves on the modular surface. These results, together with the Lindelof Hypothesis and known subconvex L∞-bounds are applied to prove that locally the number of nodal domains of such a form goes to infinity with its eigenvalue.

Real zeros of holomorphic Hecke cusp forms

This note is concerned with the zeros of holomorphic Hecke cusp forms of large weight on the modular surface. The zeros of such forms are sym- metric about three geodesic segments and we call those zeros that lie on these segments, real. Our main results give estimates for the number of real zeros as the weight goes to infinity. Mathematics Subject Classification (2010). Primary: 11F11, 11F30. Sec- ondary: 34F05.

On the zeros of the Taylor polynomials associated with the exponential function

Etude de la geometrie des zeros des polynomes de Taylor associes a la fonction exponentielle

The Number of Roots of Polynomials of Large Degree in a Prime Field

We establish asymptotic upper bounds on the number of roots modulo p of some polynomials with rational coefficients, with p an arbitrarily large prime numbers, using a variant of Stepanovs method. The polynomials we consider have degree of size close to p and are obtained by truncating power series of polylogarithms, polyexponentials, and Bessel functions. This answers a question of Heath-Brown.